scipy.linalg.qr(a, overwrite_a=False, lwork=None, mode='full')

Compute QR decomposition of a matrix.

Calculate the decomposition :lm:`A = Q R` where Q is unitary/orthogonal and R upper triangular.

Parameters :

a : array, shape (M, N)

Matrix to be decomposed

overwrite_a : bool, optional

Whether data in a is overwritten (may improve performance)

lwork : int, optional

Work array size, lwork >= a.shape[1]. If None or -1, an optimal size is computed.

mode : {‘full’, ‘r’, ‘economic’}

Determines what information is to be returned: either both Q and R (‘full’, default), only R (‘r’) or both Q and R but computed in economy-size (‘economic’, see Notes).

Returns :

Q : double or complex ndarray

Of shape (M, M), or (M, K) for mode='economic'. Not returned if mode='r'.

R : double or complex ndarray

Of shape (M, N), or (K, N) for mode='economic'. K = min(M, N).

Raises LinAlgError if decomposition fails :


This is an interface to the LAPACK routines dgeqrf, zgeqrf, dorgqr, and zungqr.

If mode=economic, the shapes of Q and R are (M, K) and (K, N) instead of (M,M) and (M,N), with K=min(M,N).


>>> from scipy import random, linalg, dot
>>> a = random.randn(9, 6)
>>> q, r = linalg.qr(a)
>>> allclose(a, dot(q, r))
>>> q.shape, r.shape
((9, 9), (9, 6))
>>> r2 = linalg.qr(a, mode='r')
>>> allclose(r, r2)
>>> q3, r3 = linalg.qr(a, mode='economic')
>>> q3.shape, r3.shape
((9, 6), (6, 6))

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